Further results on modified harmonic functions in three dimensions
The Weinstein equation tΔu+k∂u∂t=0, with k∈ℤ, considered in ℝ3=(x,y,t), is a modification of the classical Laplace equation Δu=0. Its solutions are called k‐modified harmonic functions. Whereas for positive integers k the Weinstein equation is relatively well understood, little is known if the parameter k is negative. The main result of this article is the statement that in case the negative integers are even, i.e., k=−2ℓ,ℓ∈ℕ, we still have a Fischer‐type decomposition. For k=0, the classical harmonic functions, this decomposition is well known. But also in case k∈ℕ, a Fischer‐type decomposition holds true, a Fischer‐type decomposition holds true. Surprisingly in case k=−3,k=−5, or k=−7 and probably in all higher negative odd cases, the decomposition doesn't hold. In case k=−1, we give a complete description of the vector space Hnk(ℝ3) of homogeneous k‐modified harmonic polynomials of degree n in ℝ3. Such a result is also at hand in case k∈ℕ. Finally, in case k=0 of the classical harmonic functions, we give a description of the vector space Hn(ℝ3)=Hn0(ℝ3).